GEOPHYSICS, VOL. 75, NO. 5 共SEPTEMBER-OCTOBER 2010兲; P. 75A211–75A227, 10 FIGS.
10.1190/1.3463440
Tutorial on seismic interferometry:
Part 2 — Underlying theory and new advances
Kees Wapenaar1, Evert Slob1, Roel Snieder2, and Andrew Curtis3
ABSTRACT
In the 1990s, the method of time-reversed acoustics was developed. This method exploits the fact that the acoustic wave equation for a lossless medium is invariant for time reversal. When ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the
transducers, they focus at the position of the original source, even
when the medium is very complex. In seismic interferometry the
time-reversed responses are not physically sent into the earth, but
they are convolved with other measured responses. The effect is
essentially the same: The time-reversed signals focus and create
a virtual source which radiates waves into the medium that are
subsequently recorded by receivers. A mathematical derivation,
based on reciprocity theory, formalizes this principle: The crosscorrelation of responses at two receivers, integrated over differ-
INTRODUCTION
In Part 1, we discussed the basic principles of seismic interferometry 共also known as Green’s function retrieval4兲 using mainly heuristic arguments. In Part 2, we continue our discussion, starting with an
analysis of the relation between seismic interferometry and the field
of time-reversed acoustics, pioneered by Fink 共1992, 1997兲. This
analysis includes a heuristic discussion of the virtual-source method
of Bakulin and Calvert 共2004, 2006兲 and a review of an elegant physical derivation by Derode et al. 共2003a, b兲 of Green’s function retrieval by crosscorrelation. After that, we review exact Green’s function representations for seismic interferometry in arbitrary inhomogeneous, anisotropic lossless solids 共Wapenaar, 2004兲 and discuss
the approximations that lead to the commonly used expressions. We
conclude with an overview of recent and new advances, including
ent sources, gives the Green’s function emitted by a virtual
source at the position of one of the receivers and observed by the
other receiver. This Green’s function representation for seismic
interferometry is based on the assumption that the medium is
lossless and nonmoving. Recent developments, circumventing
these assumptions, include interferometric representations for
attenuating and/or moving media, as well as unified representations for waves and diffusion phenomena, bending waves, quantum mechanical scattering, potential fields, elastodynamic, electromagnetic, poroelastic, and electroseismic waves. Significant
improvements in the quality of the retrieved Green’s functions
have been obtained with interferometry by deconvolution. A
trace-by-trace deconvolution process compensates for complex
source functions and the attenuation of the medium. Interferometry by multidimensional deconvolution also compensates for the
effects of one-sided and/or irregular illumination.
approaches that account for attenuating and/or nonreciprocal media,
methods for obtaining virtual receivers or virtual reflectors, the relationship with imaging theory, and, last but not least, interferometry
by deconvolution. The discussion of each of these advances is necessarily brief, but we include many references for further reading.
INTERFEROMETRY AND TIME-REVERSED
ACOUSTICS
Review of time-reversed acoustics
In the early 1990s, Mathias Fink and coworkers at the University
of Paris VII initiated a new field of research called time-reversed
acoustics 共Fink, 1992, 1997; Derode et al., 1995; Draeger and Fink,
Manuscript received by the Editor 30 November 2009; published online 14 September 2010.
1
Delft University of Technology, Department of Geotechnology, Delft, The Netherlands. E-mail: c.p.a.wapenaar@tudelft.nl; e.c.slob@tudelft.nl.
2
Colorado School of Mines, Center for Wave Phenomena, Golden, Colorado, U.S.A. E-mail: rsnieder@mines.edu.
3
University of Edinburgh, School of GeoSciences, Edinburgh, U.K. E-mail: andrew.curtis@ed.ac.uk.
© 2010 Society of Exploration Geophysicists. All rights reserved.
4
By “Green’s function,” we mean the response of an impulsive point source in the actual medium rather than in a background medium.
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Wapenaar et al.
positions causes the particle to follow a completely different trajectory after only a few encounters with the scatterers. Waves, on the
other hand, have a finite wavelength and travel along all possible trajectories, visiting all of the scatterers in all possible combinations.
Hence, a small perturbation in initial conditions or scatterer positions has a much less dramatic effect for wave scattering than for particle scattering. Consequently, wave-propagation experiments
through a strongly scattering medium have a high degree of repeatability. Combined with the invariance of the wave equation for time
reversal, this explains the excellent reproduction of the source wavefield after back propagation through the scattering medium.
As a historical side note, the idea of emitting time-reversed signals into a system was proposed and implemented in the 1960s 共Parvulescu, 1961, 1995兲. This was a single-channel method, aiming to
compress a complicated response at a detector 共for example, in an
ocean waveguide兲 into a single pulse. The method was proposed as a
fast alternative to digital crosscorrelation, which, with the computers at that time, cost on the order of 10 days’ computation time per
correlation for signal lengths typically considered in underwater
acoustics 共Stewart et al., 1965兲.
Snieder et al. 共2002兲 and Grêt et al. 共2006兲 exploit the repeatability
of acoustic experiments in a method they call coda wave interferometry 共here, interferometry is used in the classical sense兲. Because the
scattering coda is repeatable when an experiment is carried out twice
under the same circumstances, any change in the coda between two
experiments can be attributed to changes in the medium. As a result
of the relatively long duration of the coda, minor time-lapse changes
in, for example, the background velocity can be monitored with high
accuracy by coda wave interferometry.
Apart from repeatability, another important aspect of time-reversed acoustics is its potential to image beyond the diffraction limit.
Consider again the time-reversal experiment in Figure 1. An important effect of the scattering medium between the source at A and the
transducer array at B is a widening of the effective aperture angle. In
other words, waves that arrive at each receiver include energy from a
much wider range of take-off angles from the source location than
would be the case without scatterers. A consequence is that time-reversal experiments in strongly scattering media
have so-called superresolution properties 共de
a)
c)
Rosny and Fink, 2002; Lerosey et al., 2007兲. Ha1
nafy et al. 共2009兲 and Cao et al. 共2008兲 used this
0.5
property in a seismic time-reversal method to lo0
A
cate trapped miners accurately after a mine col–50
–25
25
50
–0.5
lapse.
–1
An essential condition for the stability and
Time (µs)
B
high-resolution aspects of time-reversed acous(First step)
b)
d)
tics is that the time-reversed waves propagate
0
through the same physical medium as in the for–5
ward experiment. Here, we see a link between
–10
time-reversed acoustics and seismic interferomeA
try. Instead of doing a real reverse-time experi–15
ment, in seismic interferometry one convolves
–20
x-axis
B
forward and time-reversed responses. Because
–25
(Second step)
both responses are measured in one and the same
–3
–2
0
2
3
–1
1
physical medium, seismic interferometry has
Distance from source (cm)
similar stability and high-resolution properties as
time-reversed acoustics. This link is made more
Figure 1. Time-reversed acoustics in a strongly scattering random medium 共Derode et al.,
1995; Fink, 2006兲. 共a兲 The source at A emits a short pulse that propagates through the ranexplicit in the next two sections.
dom medium. The scattered waves are recorded by the array at B. 共b兲 The array at B emits
Finally, note that time-reversed acoustics
the time-reversed signals, which, after back propagation through the random medium, foshould
be distinguished from reverse time migracus at A. 共c兲 The back-propagated response at A. 共d兲 Beam profiles around A.
Amplitude (dB)
Normalized amplitude
1999; Fink and Prada, 2001兲. Here, we briefly review this research
field; in the next sections, we discuss the links with seismic interferometry.
Time-reversed acoustics makes use of the fact that the acoustic
wave equation for a lossless acoustic medium is invariant under time
reversal 共because it only contains even-order time derivatives, i.e.,
zeroth and second order兲. This means that when u共x,t兲 is a solution,
then u共x,ⳮt兲 is a solution as well. Figure 1 illustrates the principle in
the context of an ultrasonic experiment 共Derode et al., 1995; Fink,
2006兲. A piezoelectric source at A in Figure 1a emits a short pulse
共duration of 1 ␮s兲 that propagates through a highly scattering medium 共a set of 2000 randomly distributed steel rods with a diameter of
0.8 mm兲. The transmitted wavefield is received by an array of piezoelectric transducers at B. The received traces 共three are shown in Figure 1a兲 exhibit a long coda 共⬎200 ␮s兲 because of multiple scattering between the rods. Next, the traces are reversed in time and simultaneously fed as source signals to the transducers at B 共Figure 1b兲.
This time-reversed wavefield propagates through the scattering medium and focuses at the position of the original source. Figure 1c
shows the received signal at the original source position; the duration is of the same order as the original signal 共⬃1 ␮s兲. Figure 1d
shows beam profiles around the source position 共amplitudes measured along the x-axis denoted in Figure 1b兲. The narrow beam is the
result of this experiment 共back propagation via the scattering medium兲, whereas the wide beam was obtained when the steel rods were
removed.
The resolution is impressive; at the time, the stability of this experiment amazed many researchers. From a numerical experiment,
one might expect such good reconstruction; however, when waves
have scattered by tens to hundreds of scatterers in a real experiment,
the fact that the wavefield refocuses at the original source point is
fascinating. Snieder and Scales 共1998兲 have analyzed this phenomenon in detail. In their analysis, they compare wave scattering with
particle scattering. They show for their model that, whereas particles
behave chaotically after having encountered typically eight scatterers, waves remain stable after 30 or more scatterers. The instability
of particle scattering is explained by the fact that particles follow a
single trajectory.Asmall disturbance in initial conditions or scatterer
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Tutorial on interferometry: Part 2
tion 共RTM兲, such as proposed by McMechan 共1982, 1983兲, Baysal et
al. 共1983兲, Whitmore 共1983兲, and Gajewski and Tessmer 共2005兲, in
which time-reversed waves are propagated numerically through a
macromodel. No matter how much detail one puts into a macromodel, a result such as the one illustrated in Figure 1 can only be obtained when the same physical medium is used in the forward as in
the reverse-time experiment. Time-reversed acoustics and reverse
time migration serve different purposes. The field of RTM has advanced significantly during the last few years, and contractors and
oil companies are now applying the method routinely for depth imaging 共Etgen et al., 2009; Zhang and Sun, 2009; Clapp et al., 2010兲.
Virtual-source method
The method of time-reversed acoustics inspired Rodney Calvert
and Andrey Bakulin at Shell to develop what they call the virtualsource method 共Bakulin and Calvert, 2004, 2006兲5.
In essence, their virtual-source method is an elegant data-driven
alternative for model-driven redatuming, similar to Schuster’s methods discussed in Part 1 共we point out the differences later兲. For an acquisition configuration with sources at the surface and receivers in
the subsurface — for example, in a near-horizontal borehole 共Figure
2兲 — the reflection response is described as u共xB,x共i兲
S ,t兲 ⳱ G共x B,
共i兲
x共i兲
,t兲
ⴱ
s共t兲,
where
s共t兲
is
the
source
wavelet
and
G共x
B,x S ,t兲 the
S
Green’s function, describing propagation from a point source at x共i兲
S
via a target below the borehole to a receiver at xB in the borehole 共we
adopt the notation of Part 1; the asterisk denotes temporal convolution兲. The downgoing wavefield observed by a downhole receiver at
共i兲
xA is given by u共xA,x共i兲
S ,t兲 ⳱ G共x A,x S ,t兲 ⴱ s共t兲. Using source-receiver
共i兲
reciprocity, i.e., u共xA,x共i兲
,t兲
⳱
u共x
,x
A,t兲, this can also be interpreted
S
S
as the response of a downhole source at xA, observed by an array of
receivers x共i兲
S after propagation through the complex overburden.
This response is comparable with the response of the ultrasonic experiment in Figure 1a. Hence, if all traces u共x共i兲
S ,x A,t兲 would be reversed in time and fed simultaneously as source signals to the sources at x共i兲
S , similar as in Figure 1b, the back-propagating wavefield
would focus at xA. Instead of doing this physically, the time-reversed
signals are convolved with the reflection responses and subsequently summed over the different source positions at the surface according to
共i兲
C共xB,xA,t兲 ⳱ 兺 u共xB,x共i兲
S ,t兲 ⴱ u共xA,xS ,ⳮt兲.
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gate through the strongly scattering medium and focus to a short-duration pulse at A, in Bakulin and Calvert’s method the sources at the
surface are focused to a virtual source in the borehole, compensating
for a complex overburden. Similar to the time-reversed acoustics
method, the focusing occurs with a time-reversed measured response; hence, the redatuming takes place in the same physical medium as the one in which the data were measured. This distinguishes
the virtual-source method from classical redatuming 共Berryhill,
1979, 1984兲 and the common-focal-point 共CFP兲 method 共Berkhout,
1997; Berkhout and Verschuur, 2001兲. Each of these methodologies
has its own applications and hence its own right of existence. Classical redatuming and the CFP method are applied to data acquired by
sources and receivers at the surface, using as operators either modelbased Green’s functions 共redatuming兲 or dynamic focusing operators that are aimed to converge iteratively to the Green’s functions
共CFP method兲. The virtual-source method uses sources at the surface
and receivers in a borehole that directly measure the operators. The
idea of using measured Green’s functions as redatuming operators
may seem simple with hindsight, but the consequences are far reaching. Bakulin et al. 共2007兲 give an impressive overview of the applications in imaging and reservoir monitoring.
A new method for wavelet estimation has been proposed as an interesting corollary of the virtual-source method 共Behura, 2007兲.
When the virtual source coincides with a real source at xA, the response at xB from the real source is given by G共xB,xA,t兲 ⴱ s共t兲. The
virtual-source response, obtained by equation 1, is given by
G共xB,xA,t兲 ⴱ Ss共t兲, with Ss共t兲 ⳱ s共t兲 ⴱ s共ⳮt兲. Hence, deconvolution
of the virtual-source response by the actual response gives the 共timereversed兲 wavelet.
Last but not least, we remark that an important difference of equation 1 with the previously discussed expressions for seismic interferometry in Part 1 is the single-sidedness of the correlation function
C共xB,xA,t兲 ⬇ G共xB,xA,t兲 ⴱ Ss共t兲 关there is no time-reversed term
G共xB,xA,ⳮt兲兴. Moreover, this correlation function is only approximately proportional to the causal Green’s function. These are consequences of the anisotropic illumination of the receivers in the borehole, which are primarily illuminated from above. In the “Acoustic
Representation” section, we revisit the approximations of one-sided
(i)
共1兲
xS
i
(i)
The correlation function C共xB,xA,t兲 is interpreted as the response of a
virtual downhole source at xA, measured by a downhole receiver at
xB; hence, C共xB,xA,t兲 ⬇ G共xB,xA,t兲 ⴱ Ss共t兲. The wavelet of the virtual
source, Ss共t兲, is the autocorrelation of the wavelet s共t兲 of the real
sources at the acquisition surface. Similar to Schuster’s methods,
equation 1 can be seen as a form of source redatuming, using a
measured version of the redatuming operator, i.e., u共xA,x共i兲
S ,ⳮt兲
⳱ G共xA,x共i兲
S ,ⳮt兲 ⴱ s共ⳮt兲.
Whereas in Schuster’s methods the emphasis is on aspects such as
transforming multiples into primaries, enlarging the illumination
area, and interpolating missing traces, the emphasis of Bakulin and
Calvert’s virtual-source method is on eliminating the propagation
distortions of the complex inhomogeneous overburden. Similar to
Figure 1, where the time-reversed complex signals at B back propa-
u(xA ,xS ,t)
Complex
overburden
xA ( Virtual source)
xB
Well
G(xB ,xA ,t)
(i)
u(xB ,xS ,t)
Simpler
middle
overburden
Target
Figure 2. Basic principle of the virtual-source method of Bakulin
and Calvert 共2004, 2006兲. Receivers in a borehole record the downgoing wavefield through the complex overburden and the reflected
signal from the deeper target. Crosscorrelation and summing over
source locations give the reflection response of a virtual source in the
borehole, free of overburden distortions.
5
Recall from part 1 that creating a virtual source is the essence of all seismic interferometry methods; hence, we use the term virtual source when appropriate.
When it refers to Bakulin and Calvert’s method, we mention this explicitly 共unless when it is clear from the context兲.
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75A214
Wapenaar et al.
illumination and indicate various improvements. The most effective
improvement is discussed in the “Interferometry by multidimensional deconvolution” section.
Derivation of seismic interferometry from time-reversed
acoustics
The virtual-source method, although very elegant, is an intuitive
application of time-reversed acoustics. Derode et al. 共2003a, b兲 show
more precisely how the principle of Green’s function retrieval by
crosscorrelation in open systems can be derived from time-reversed
acoustics. Their derivation is based entirely on physical arguments
and shows that Green’s function retrieval 共which is equivalent to
seismic interferometry兲, holds for arbitrarily inhomogeneous lossless media, including highly scattering media as shown in Figure 1.
Here, we briefly review their arguments, but we replace their notation by that used in Part 1.
Consider a lossless arbitrary inhomogeneous acoustic medium in
a homogeneous embedding. In this configuration, we define two
points with coordinate vectors xA and xB. Our aim is to show that the
acoustic response at xB from an impulsive source at xA 共i.e., the
Green’s function G共xB,xA,t兲兲 can be obtained by crosscorrelating ob-
a)
x
b)
x'
⬀
x
G( x, xA ,– t )
c)
G (x B ,x A ,t )
G( xA ,x,t )
共3兲
G共xB,xA,t兲 Ⳮ G共xB,xA,ⳮt兲
G(x', x,t )
xA
where ⬀ denotes “proportional to.” According to equation 2, G共x⬘,
x,t兲 propagates the source function G共x,xA,ⳮt兲 from x to x⬘ and the
result is integrated over all sources on ⳵D. Because of the invariance
of the acoustic wave equation for time-reversal, we know that the
wavefield u共x⬘,t兲 must focus at x⬘ ⳱ xA and t ⳱ 0. This property is
the basis of time-reversed acoustics and explains why the focusing in
Figure 1 occurs.
Derode et al. 共2003a, b兲 go one step further in their interpretation
of equation 2. Because u共x⬘,t兲 focuses for x⬘ ⳱ xA at t ⳱ 0, the wavefield u共x⬘,t兲 for arbitrary x⬘ and t can be seen as the response of a virtual source at xA and t ⳱ 0. This virtual-source response, however,
consists of a causal part and an acausal part, according to
This expression is explained as follows: The wavefield generated by
the acausal sources on ⳵D first propagates to all x⬘ where it gives an
acausal contribution; next, it focuses in xA at t ⳱ 0. Finally, because
the energy focused at that point is not extracted from the system, it
must propagate outward again to all x⬘, giving the causal contribution. The propagation paths from x⬘ to xA are the same as those from
xA to x⬘ but are traveled in the opposite direction, which explains the
time-symmetric form of u共x⬘,t兲.
Combining equations 2 and 3, applying source-receiver reciprocity to G共x,xA,ⳮt兲 in equation 2, and setting x⬘ ⳱ xB yields
G( x, xA ,t )
xA
共2兲
u共x⬘,t兲 ⳱ G共x⬘,xA,t兲 Ⳮ G共x⬘,xA,ⳮt兲.
x'
xA
servations of wavefields at xA and xB due to sources on a closed surface ⳵D in the homogeneous embedding. The derivation starts by
considering another experiment: an impulsive source at t ⳱ 0 at xA
and receivers at x on ⳵D 共Figure 3a兲. The response at any point x on
⳵D is denoted by G共x,xA,t兲. Imagine that we record this response for
all x on ⳵D, reverse the time axis, and simultaneously feed these
time-reversed functions G共x,xA,ⳮt兲 to sources at all positions x on
⳵D 共Figure 3b兲. The superposition principle states that the wavefield
at any point x⬘ inside ⳵D due to these sources on ⳵D is given by
xB
G (x B ,x , t)
x
Figure 3. Derivation of Green’s function retrieval, using arguments
from time-reversed acoustics 共Derode et al., 2003a, b兲. 共a兲 Response
of a source at xA, observed at any x 共the ray represents the full response, including primary and multiple scattering due to inhomogeneities兲. 共b兲 The time-reversed responses are emitted back into the
medium. 共c兲 The response of a virtual source at xA can be obtained
from the crosscorrelation of observations at two receivers and integration along the sources.
冖
⳵D
G共xB,x,t兲 ⴱ G共xA,x,ⳮt兲d2x.
共4兲
We recognize the now well-known form of an interferometric relation with, on the left-hand side, the Green’s function between xA and
xB plus its time-reversed version and, on the right-hand side, crosscorrelations of wavefield observations at xA and xB, integrated along
the sources at x on ⳵D 共Figure 3c兲. The right-hand side can be reduced to a single crosscorrelation of noise observations in a similar
way as discussed in Part 1 共we briefly review this later in the section
“Acoustic representation”兲.
Note that equation 4 holds for an arbitrarily inhomogeneous medium inside ⳵D; hence, the reconstructed Green’s function
G共xB,xA,t兲 contains the ballistic wave 共i.e., the direct wave兲 as well
as the coda resulting from multiple scattering in the inhomogeneous
medium. In itself, this is not new because equation 13 in Part 1 is also
derived for inhomogeneous media. However, equation 4 is derived
directly from the principle of time-reversed acoustics, so it now follows that seismic interferometry has the same favorable stability and
resolution properties as time-reversed acoustics. Sens-Schönfelder
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Tutorial on interferometry: Part 2
and Wegler 共2006兲 and Brenguier et al. 共2008b兲 exploit the stability
properties by applying coda wave interferometry 共Snieder et al.,
2002兲 to Green’s functions obtained by crosscorrelating noise observations at different seismometers on a volcano. With this method,
they can measure velocity variations with an accuracy of 0.1% with a
temporal resolution of a single day. Brenguier et al. 共2008a兲 use a
similar method to monitor changes in seismic velocity associated
with earthquakes near Parkfield, California.
The derivation of Derode et al. 共2003a, b兲 that we have reviewed
here is entirely based on elegant physical arguments, but it is not
mathematically exact. In the next section, we derive exact expressions and show the approximations that need to be made to arrive at
equation 4.
GREEN’S FUNCTION REPRESENTATIONS FOR
SEISMIC INTERFEROMETRY
Equations 13 and 14 in Part 1 express the reflection response of a
3D inhomogeneous medium in terms of crosscorrelations of the
transmission responses of that medium. We derived these relations
in 2002 as a generalization of Claerbout’s 1D expressions 共equations
8 and 12 in Part 1兲. The derivation was based on a reciprocity theorem of the correlation type for one-way wavefields. To establish a
link with the independently upcoming field of Green’s function retrieval, in 2004 we derived the equivalent of these relations in terms
of Green’s functions for full wavefields 共Wapenaar, 2004兲. The starting point was the Rayleigh-Betti reciprocity theorem for elastodynamic wavefields. Apart from establishing the mentioned link, this
derivation has the additional advantage that the inherent approximations of the one-way reciprocity theorem of the correlation type are
circumvented 共or at least postponed to a later stage in the derivation兲.
Elastodynamic representation
Here, we briefly review our derivation of the elastodynamic
Green’s function representation for interferometry and discuss the
connection with the methods discussed in the previous section and in
Part 1. Consider an arbitrarily heterogeneous and anisotropic lossless solid medium with stiffness cijkl共x兲 and mass density ␳ 共x兲. In
this medium, an external force distribution f i共x,t兲 generates an elastodynamic wavefield, characterized by stress tensor ␶ ij共x,t兲 and particle velocity vi共x,t兲. The Fourier transforms of these time-dependent quantities are defined via
f̂共␻ 兲 ⳱
冕
⬁
ⳮ⬁
f共t兲exp共ⳮj␻ t兲dt,
共5兲
where ␻ is the angular frequency and j is the imaginary unit. In the
space-frequency domain, the stress-strain relation reads j␻ ␶ˆ ij
ⳮ cijkl⳵lv̂k ⳱0 and the equation of motion is j␻␳ v̂i ⳮ ⳵ j␶ˆ ij ⳱ f̂ i. The
operator ⳵ j denotes the partial derivative in the x j-direction, and Einstein’s summation convention applies to repeated subscripts.
In the following, we consider two independent elastodynamic
states 共i.e., sources and wavefields兲 distinguished by subscripts A
and B. For an arbitrary spatial domain D enclosed by boundary ⳵D
with outward-pointing normal n ⳱ 共n1,n2,n3兲, the Rayleigh-Betti
reciprocity theorem that relates these two states is given by
冕
75A215
兵f̂ i,Av̂i,B ⳮ v̂i,A f̂ i,B其d3x ⳱
D
冖
⳵D
兵v̂i,A␶ˆ ij,B ⳮ ␶ˆ ij,Av̂i,B其n jd2x
共6兲
共Knopoff and Gangi, 1959; de Hoop, 1966; Aki and Richards, 1980兲.
This theorem is also known as a reciprocity theorem of the convolution type because all products in the frequency domain, such as
v̂i,A␶ˆ ij,B, correspond to convolutions in the time domain.
Similarly to the acoustic situation, we can apply the principle of
time-reversal invariance for elastic waves in a lossless medium 共Bojarski, 1983兲. Time reversal corresponds to complex conjugation in
the frequency domain. Hence, when stress tensor ␶ˆ ij and particle velocity v̂i are solutions of the stress-strain relation and the equation of
motion with source term f̂ i, then ␶ˆ ij* and ⳮ v̂i* obey the same equations with source term f̂ i* 共the negative sign in ⳮ v̂i* comes from the
replacement 共 j␻ 兲* ⳱ ⳮj␻ in the equation of motion兲. Making these
substitutions for state A, we obtain
冕
D
* v̂ Ⳮ v̂* f̂ 其d3x
兵f̂ i,A
i,B
i,A i,B
⳱
冖
⳵D
2
* ␶ˆ
*
兵ⳮ v̂i,A
ij,B ⳮ ␶ˆ ij,Av̂i,B其n jd x.
共7兲
This is an elastic reciprocity theorem of the correlation type because
* ˆ
products such as v̂i,A
␶ ij,B correspond to correlations in the time domain.
Next, we replace the wavefields in both states in equation 7 by
Green’s functions. This means that we replace the force distributions
by unidirectional impulsive point forces in both states, according to
f i,A共x,t兲 ⳱ ␦ 共x ⳮ xA兲␦ 共t兲␦ ip and f i,B共x,t兲 ⳱ ␦ 共x ⳮ xB兲␦ 共t兲␦ iq in the
time domain or f̂ i,A共x,␻ 兲 ⳱ ␦ 共x ⳮ xA兲␦ ip and f̂ i,B共x,␻ 兲 ⳱ ␦ 共x
ⳮ xB兲␦ iq in the frequency domain, with xA and xB in D and where indices p and q denote the directions of the applied forces. Accordingly, for the particle velocities, we substitute v̂i,A共x,␻ 兲 ⳱ Ĝip共x,xA,␻ 兲
and v̂i,B共x,␻ 兲 ⳱ Ĝiq共x,xB,␻ 兲, respectively. Here, Ĝip共x,xA,␻ 兲 represents the i component of the particle velocity at x due to a unit force
source in the p direction at xA, etc. Substituting these sources and
Green’s functions into equation 7, using the stress-strain relation and
source-receiver reciprocity 共i.e., Ĝip共x,xA,␻ 兲 ⳱ Ĝ pi共xA,x,␻ 兲兲, gives
* 共x ,x ,␻ 兲
Ĝqp共xB,xA,␻ 兲 Ⳮ Ĝqp
B A
⳱ⳮ
冖
cijkl共x兲
共共⳵lĜqk共xB,x,␻ 兲兲Ĝ*pi共xA,x,␻ 兲
j
␻
⳵D
ⳮ Ĝqi共xB,x,␻ 兲⳵lĜ*pk共xA,x,␻ 兲兲n jd2x.
共8兲
In the time domain,
⳵t 兵Gqp共xB,xA,t兲 Ⳮ Gqp共xB,xA,ⳮ t兲其
⳱ⳮ
冖
⳵D
cijkl共x兲共⳵lGqk共xB,x,t兲 ⴱ G pi共xA,x,ⳮt兲
ⳮ Gqi共xB,x,t兲 ⴱ ⳵lG pk共xA,x,ⳮt兲兲n jd2x.
共9兲
Note that this representation has a similar form as many of the expressions we have encountered before. It is an exact representation
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for the Green’s function between xA and xB plus its time-reversed
version, expressed in terms of crosscorrelations of wavefield observations at xA and xB, integrated along the sources at x on ⳵D. It holds
for an arbitrarily inhomogeneous anisotropic medium 共inside as well
as outside ⳵D兲, and the closed boundary ⳵D containing the sources of
the Green’s functions may have any shape. When a part ⳵D0 of the
boundary is a stress-free surface, as in Figure 4, then the integrand of
the right-hand side of equation 7 is zero on ⳵D0. Consequently, the
boundary integral in equation 9 need only be evaluated over the remaining part ⳵D1 共meaning that sources can be restricted to that part
of the boundary兲. Note that equation 9 still holds in the limiting case
in which xA and xB lie at the free surface. In that case, the Green’s
functions on the left-hand side have a traction source at xA 共Wapenaar and Fokkema, 2006兲.
An important difference with earlier expressions is that the righthand side of representation 9 contains a combination of two terms,
where each of the terms is a crosscorrelation of Green’s functions
with different types of sources at x 共e.g., the operator ⳵l in
⳵lGqk共xB,x,t兲 is a differentiation with respect to xl, which changes the
character of the source at x of this Green’s function兲. For modeling
applications, this is not a problem because any type of source can be
defined in modeling. Van Manen et al. 共2006, 2007兲 use equation 9
for what they call interferometric modeling. They model the response of different types of sources on a boundary and save the responses for all possible receiver positions in the volume enclosed by
the boundary. Next, they apply equation 9 to obtain the responses of
all possible source positions in that volume. Hence, for the cost of
modeling responses of sources on a boundary 共and calculating many
crosscorrelations兲, they obtain responses of sources throughout a
volume. This approach can be very useful for nonlinear inversion
schemes, where, in each iteration, Green’s functions for sources in a
volume are required.
The requirement of correlating responses of different types of
sources makes equation 9 in its present form less practical for application in seismic interferometry. This is particularly true for passive
data, where one must rely on the availability of natural sources. To
accommodate this situation, equation 9 can be modified 共Wapenaar
and Fokkema, 2006兲. Here, we only indicate the main steps. Using a
D0
xA
G qp (x B ,xA ,t )
xB
x
D1
Figure 4. Configuration for elastodynamic Green’s function retrieval 共the rays represent the full response, including primary and multiple scattering as well as mode conversion due to inhomogeneities兲.
In this configuration, a part of the closed boundary is a free surface
共⳵D0兲, so sources are only required on the remaining part of the
boundary 共⳵D1兲. The shallow sources 共say, above the dashed line兲 are
mainly responsible for retrieving the surface waves and the direct
and shallowly refracted waves in Gqp共xB,xA,t兲, whereas the deeper
sources mainly contribute to the retrieval of the reflected waves in
Gqp共xB,xA,t兲.
high-frequency approximation, assuming the medium outside ⳵D is
homogeneous and isotropic, the sources can be decomposed into Pand S-wave sources and their derivatives in the direction of the normal on ⳵D. These derivatives can be approximated, leading to a simplified version of equation 9 in which only crosscorrelations of
Green’s functions with the same source type occur. This approximation is accurate when ⳵D is a sphere with large radius. It can also be
used for arbitrary surfaces ⳵D but at the expense of amplitude errors.
Because the approximation does not affect the phase, it is usually
considered acceptable for seismic interferometry. Finally, when the
sources are mutually uncorrelated noise sources for P- and S-waves
on ⳵D, equation 9 reduces to
兵Gqp共xB,xA,t兲 Ⳮ Gqp共xB,xA,ⳮt兲其 ⴱ SN共t兲
⬇
2
具vq共xB,t兲 ⴱ v p共xA,ⳮt兲典,
␳ cP
共10兲
where v p共xA,t兲 and vq共xB,t兲 are the p and q components of the particle velocity of the noise responses at xA and xB, respectively; SN共t兲 is
the autocorrelation of the noise; and cP is the P-wave propagation velocity of the homogeneous medium outside ⳵D. For the configuration of Figure 4, the Green’s function Gqp共xB,xA,t兲 retrieved by equation 10 contains the surface waves between xA and xB as well as the
reflected and refracted waves, assuming the noise sources are well
distributed over the source boundary ⳵D1 in the half-space below the
free surface. In practice, equation 10 is used either for surface-wave
or for reflected-wave interferometry.
For surface-wave interferometry, the sources at and close to the
surface typically give the most relevant contributions — say, the
sources above the dashed line in Figure 4. In our earlier, more intuitive discussions on direct-wave interferometry in Part 1, we consider
the fundamental surface-wave mode as an approximate solution of a
2D wave equation in the horizontal plane and argue that the Green’s
function of this fundamental mode can be extracted by crosscorrelating ambient noise. Equation 10 is a corollary of the exact 3D representation 9 and thus accounts not only for the fundamental mode of
the direct surface wave but also for higher-order modes as well as for
scattered surface waves. Halliday and Curtis 共2008兲 carefully analyze the contributions of the different sources to the retrieval of surface waves. They show that when only sources at the surface are
available, there is strong spurious interference between higher
modes and the fundamental mode, whereas the presence of sources
at depth 共between the free surface and, say, the dashed line in Figure
4兲 enables the correct recovery of all modes independently. Nevertheless, they show that it is possible to obtain the latter result using
only surface sources if modes are separated before crosscorrelation,
are correlated separately, and are reassembled thereafter. Kimman
and Trampert 共2010兲 show that the spurious interference is also suppressed when the surface sources are very far away or organized in a
band. Halliday and Curtis 共2009b兲 analyze the requirements in terms
of source distribution for the retrieval of scattered surface waves.
Halliday et al. 共2010a兲 use the acquired insights to remove scattered
surface waves 共ground roll兲 from seismic shot records 共Figure 5兲.
For reflected-wave interferometry, the deeper situated sources
共typically those below the dashed line in Figure 4兲 give the main contributions. This is in agreement with our discussion on the retrieval
of the 3D reflection response from transmission data, for which we
consider a configuration with sources in the lower half-space 共Figure
12, Part 1兲. For this configuration, Green’s function representations
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Tutorial on interferometry: Part 2
9 and 10 can be seen as alternatives for reflection representations 13
and 14 in Part 1, generalized for an anisotropic solid medium.
Acoustic representation
Starting with Rayleigh’s reciprocity theorem 共Rayleigh, 1878; de
Hoop, 1988; Fokkema and van den Berg, 1993兲 and the principle of
time-reversal invariance 共Bojarski, 1983; Fink, 1992兲, we obtain the
acoustic analogue of equation 8 according to
Ĝ共xB,xA,␻ 兲 Ⳮ Ĝ*共xB,xA,␻ 兲
冖
⳱ⳮ
1
共共⳵iĜ共xB,x,␻ 兲兲Ĝ*共xA,x,␻ 兲
⳵D j ␻␳ 共x兲
ⳮ Ĝ共xB,x,␻ 兲⳵iĜ*共xA,x,␻ 兲兲nid2x
共11兲
共van Manen et al., 2005; Wapenaar et al., 2005兲. Here, Ĝ共xA,x,␻ 兲
⳱ Ĝ共x,xA,␻ 兲 is a solution of the wave equation
␳ ⳵i共␳ ⳮ1⳵iĜ兲 Ⳮ
冉 冊
␻2
Ĝ ⳱ ⳮj␻␳ ␦ 共x ⳮ xA兲,
c2
共12兲
for an arbitrarily inhomogeneous lossless fluid medium with propagation velocity c ⳱ c共x兲 and mass density ␳ ⳱ ␳ 共x兲.
Before we discuss its use in seismic interferometry, we remark
that equation 11 has been used in almost the same form in optical holography 共Porter, 1970兲, seismic migration 共Esmersoy and Oristaglio, 1988兲, and acoustic inverse scattering 共Oristaglio, 1989兲 共except that in those papers the Green’s functions are defined without
the factor j␻␳ in the right-hand side of equation 12, leading to a
somewhat different form of equation 11兲. In imaging and inversion
literature, Ĝ共xB,xA,␻ 兲 Ⳮ Ĝ*共xB,xA,␻ 兲 is also called the homogeneous Green’s function because Ĝh共x,xA,␻ 兲 ⳱ Ĝ共x,xA,␻ 兲
Ⳮ Ĝ*共x,xA,␻ 兲 obeys the homogeneous wave equation
␳ ⳵i共␳ ⳮ1⳵iĜh兲 Ⳮ
冉 冊
␻2
Ĝh ⳱ 0
c2
共“homogeneous” means source free in this context兲. The homogeneous Green’s function
Ĝh共x,xA,␻ 兲 can also be seen as the resolution
function of the imaging integral. For a homogeneous medium, it is given by
⳱ ␻␳
冉
e jkr
eⳮjkr
ⳮ
4␲ r
4␲ r
sin共kr兲
,
2␲ r
冊
共14兲
with k ⳱ ␻ / c and r ⳱ 兩x ⳮ xA兩. This function has
its maximum for r → 0, where the amplitude is
equal to ␻ 2␳ / 2␲ c. The width of the main lobe
共measured at the zero crossings兲 is equal to the
wavelength ␭ ⳱ 2␲ / k. For a further discussion
on the relation between seismic interferometry
and the migration resolution integral, see van
Manen et al. 共2006兲, Thorbecke and Wapenaar
共2007兲, and Halliday and Curtis 共2010兲.
Consider again the acoustic Green’s function representation for
seismic interferometry 共equation 11兲. Note that in comparison with,
for example, equation 4, the right-hand side contains a combination
of two terms, where each term is a crosscorrelation of Green’s functions with different types of sources 共monopoles and dipoles兲 at x.
Here, we discuss in more detail how we can combine the two correlation products in equation 11 into a single term. To this end, we assume that the medium outside ⳵D is homogeneous, with constant
propagation velocity c and mass density ␳ . In the high-frequency regime, the derivatives of the Green’s functions can be approximated
by multiplying each constituent 共direct wave, scattered wave, etc.兲
by ⳮjk兩cos ␣ 兩, where ␣ is the angle between the relevant ray and the
normal on ⳵D. The main contributions to the integral in equation 11
come from stationary points on ⳵D.At those points, the ray angles for
both Green’s functions are identical 共see Appendix A of Part 1兲. This
implies that the contributions of the two terms under the integral in
equation 11 are approximately equal 共but opposite in sign兲; hence,
Ĝ共xB,xA,␻ 兲 Ⳮ Ĝ*共xB,xA,␻ 兲
⬇ⳮ
2
j␻␳
冖
⳵D
共ni⳵iĜ共xB,x,␻ 兲兲Ĝ*共xA,x,␻ 兲d2x.
共15兲
The integrand contains a single crosscorrelation product of dipole
and monopole source responses. When only monopole responses are
available, the operation ni⳵i can be replaced by a pseudodifferential
operator acting along ⳵D or by multiplications with ⳮ jk兩cos ␣ 兩 at
the stationary points when the ray angles are known. Hence, for controlled-source interferometry, in which case the source positions are
known and ⳵D is a smooth surface, equation 15 is a useful expression.
In passive interferometry, the positions of the sources are unknown and ⳵D can be very irregular. In that case, the best one can do
is to replace the operation ni⳵i by a factor ⳮ jk, which leads to
共13兲
a)
Time (s)
Ĝh共x,xA,␻ 兲 ⳱ j␻␳
75A217
b)
c)
0.5
0.5
0.5
1.0
1.0
1.0
1.5
1.5
1.5
2.0
2.0
2.0
2.5
2.5
2.5
–1000 –500 0 500 1000
Offset (m)
–1000 –500 0 500 1000
Offset (m)
–1000 –500 0 500 1000
Offset (m)
Figure 5. Example of interferometric ground-roll removal applied to shot records while
preserving the direct ground roll 共Halliday et al., 2010a兲. 共a兲 Raw data. 共b兲 Results of interferometric scattered ground roll removal. 共c兲 The subtracted scattered ground roll.
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Wapenaar et al.
兵G共xB,xA,t兲 Ⳮ G共xB,xA,ⳮ t兲其 ⴱ SN共t兲
Ĝ共xB,xA,␻ 兲 Ⳮ Ĝ*共xB,xA,␻ 兲
⬇
2
␳c
冖
⳵D
Ĝ共xB,x,␻ 兲Ĝ*共xA,x,␻ 兲d2x.
共16兲
This approximation is accurate when ⳵D is a sphere with large radius
so that all rays are approximately normal to ⳵D 共i.e., ␣ ⬇ 0兲. For arbitrary surfaces, equation 16 involves an amplitude error. Moreover,
spurious events may occur due to incomplete cancellation of contributions from different stationary points. However, because the approximation does not affect the phase, equation 16 is usually considered acceptable for seismic interferometry. Transforming both sides
of equation 16 back to the time domain yields
G共xB,xA,t兲 Ⳮ G共xB,xA,ⳮt兲
⬇
2
␳c
冖
⳵D
G共xB,x,t兲 ⴱ G共xA,x,ⳮt兲d2x,
共17兲
which is equal to equation 4, i.e., the expression obtained by Derode
et al. 共2003a, b兲 with proportionality factor 2 / ␳ c.
Of course, there are situations for which the derivation presented
above does not apply. For example, when ⳵D is enclosing the water
layer for marine seismology applications, the assumption that the
medium is homogeneous outside ⳵D breaks down and the derivatives of the Green’s functions must be obtained in another way.
Ramírez and Weglein 共2009兲 discuss a correlation-based processing
scheme for ocean-bottom data, based on a variant of equation 11, in
which the time-reversed Green’s function and its derivative are taken as analytic direct-wave solutions in the water layer. In the following, we restrict the application of equations 15–17 to situations for
which they were derived.
The practical application of equations 11 and 15–17 requires discretization of the integrals. The accuracy depends on the regularity
of the distribution of the sources along ⳵D 共van Manen et al., 2005;
Fan and Snieder, 2009; Yao and van der Hilst, 2009兲. A bias can be
introduced in Green’s function estimates when amplitudes of energy
have directional variations. Curtis and Halliday 共2010a兲 present an
algorithm to remove this bias. In the “Interferometry by multidimensional deconvolution” section, we present another effective way to
compensate for illumination irregularities.
Equations 11 and 15–17 have been used for interferometric wavefield modeling 共van Manen et al., 2005兲 as well as for the derivation
of passive and controlled-source seismic interferometry. For passive
interferometry, the configuration is chosen similarly to Figure 4, in
which a part of the closed boundary ⳵D is a free surface at which no
sources are required; thus, the closed boundary integral reduces to
an integral over the remaining part ⳵D1. When the sources on ⳵D1
are noise sources, the responses at xA and xB are given by u共xA,t兲
⳱ 兰⳵D1G共xA,x,t兲ⴱN共x,t兲d2x and u共xB,t兲 ⳱ 兰⳵D1G共xB,x⬘,t兲ⴱN共x⬘,
t兲d2x⬘, respectively. Assuming the noise sources are mutually uncorrelated, according to 具N共x⬘,t兲 ⴱ N共x,ⳮt兲典 ⳱ ␦ 共x ⳮ x⬘兲SN共t兲 for x
and x⬘ on ⳵D1, the crosscorrelation of the responses at xA and xB gives
具u共xB,t兲 ⴱ u共xA,ⳮt兲典
⳱
冕
⳵D1
G共xB,x,t兲 ⴱ G共xA,x,ⳮt兲 ⴱ SN共t兲d2x.
Combining this with equation 17, we obtain
共18兲
⬇
2
具u共xB,t兲 ⴱ u共xA,ⳮ t兲典.
␳c
共19兲
Representations 17 and 19 can be seen as alternatives for equations
13 and 14 in Part 1. The main difference is that, in the present derivation, we did not need to neglect evanescent waves; the receiver positions xA and xB can be anywhere in D 共instead of at the free surface兲.
Schuster 共2009兲 uses equations 11 and 15–17 for the theoretical
justification of controlled-source interferometry in any of the configurations in Figure 11 of Part 1. Korneev and Bakulin 共2006兲 use
the same equations to explain the theory of the virtual-source method illustrated in Figure 2 of the current paper. In none of these configurations do the sources form a closed boundary around the receivers at xA and xB, as prescribed by the theory, so the closed boundary
integral is by necessity replaced by an open boundary integral. Assuming the medium is sufficiently inhomogeneous such that all energy is scattered back to the receivers, one-sided illumination suffices
共Wapenaar, 2006a兲. However, in many practical situations, this condition is not fulfilled; so the open boundary integral introduces artifacts, often denoted as spurious multiples 共Snieder et al., 2006b兲.
A partial solution, implemented by Bakulin and Calvert 共2006兲, is
to apply a time window to G共xA,x,t兲 in equation 17 共or u共xA,x共i兲
S ,t兲 in
equation 1兲 with the aim of selecting direct waves only. The artifacts
can be further suppressed by applying up/down decomposition to
both Green’s functions in the right-hand side of equation 17 共Mehta
et al., 2007a; van der Neut and Wapenaar, 2009兲. Note that in the latter two cases, the direct-wave part of G共xA,x,t兲 共or u共xA,x共i兲
S ,t兲 in
equation 1兲 propagates only through the overburden. This implies
that the condition of having a lossless medium only applies to the
overburden; hence, the medium below the receivers in Figure 2 may
be attenuating. This is shown more rigorously by Slob and Wapenaar
共2007a兲 and Vasconcelos et al. 共2009兲. An even more effective suppression of artifacts related to one-sided illumination is discussed in
the “Interferometry by multidimensional deconvolution” section.
RECENT AND NEW ADVANCES
The previous discussion covers the current state of seismic interferometry. Here, we briefly indicate some recent and new advances.
Media with losses
Until now, we generally assumed that the medium under investigation is lossless and nonmoving, which is equivalent to assuming
that the underlying wave equation is invariant for time reversal.
Moreover, in all cases, the Green’s functions obey source-receiver
reciprocity. In a medium with losses, the wave equation is no longer
invariant for time reversal; but as long as the medium is not moving,
source-receiver reciprocity still holds. When the losses are not too
high, the methods discussed above yield a Green’s function with correct traveltimes and approximate amplitudes 共Roux et al., 2005;
Slob and Wapenaar, 2007b兲.
Snieder 共2007兲 shows that when the losses are significant, a
volume integral ⳮ2␻ 兰D␬ˆ i共x,␻ 兲Ĝ共xB,x,␻ 兲Ĝ*共xA,x,␻ 兲d3x 共where
␬ˆ i共x,␻ 兲 denotes the imaginary part of the compressibility兲 should be
added to the right-hand side of any of equations 11, 15 or 16 共actually, the negative sign in front of the integral is absent in Snieder’s
analysis because he uses another convention for the Fourier transform兲. Thus, in addition to the requirement of having sources at the
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Tutorial on interferometry: Part 2
boundary ⳵D 共as in Figures 3c and 4兲, sources are required throughout the domain D. When these sources are uncorrelated noise sources, the final expression for Green’s function retrieval again has a similar form as equation 19. This volume integral approach to Green’s
function retrieval is not restricted to acoustic waves in lossy media
but also applies to electromagnetic waves in conductive media 共Slob
and Wapenaar, 2007a兲 as well as to pure diffusion phenomena
共Snieder, 2006兲.
In most practical situations, sources are not available throughout a
volume. Interferometry by crossconvolution 共Slob et al., 2007a;
Halliday and Curtis, 2009b兲 is another approach that accounts for
losses. Draganov et al. 共2010兲 compensate for losses with an inverse
attenuation filter. By doing this adaptively 共aiming to minimize artifacts兲, they estimate the attenuation parameters. The methodology
discussed in the “Interferometry by multidimensional deconvolution” section also accounts very effectively for losses.
Nonreciprocal media
In a moving medium 共with or without losses兲, time-reversal invariance and source-receiver reciprocity break down. It has been
shown that with some modifications, time-reversed acoustic focusing 共as in Figure 1兲 can still work in a moving medium 共Dowling,
1993; Roux et al., 2004兲. Using reciprocity theory, it follows that
Green’s function retrieval by crosscorrelation is also possible in a
moving medium 共Godin, 2006; Wapenaar, 2006b兲. The required
modification to the Green’s function representation is surprisingly
simple: The time-reversed Green’s function G共xB,xA,ⳮt兲 on the lefthand side of equations 17 and 19 should be replaced by G共xA,xB,ⳮt兲,
assuming all Green’s functions appearing in the representation are
defined in the moving medium. Hence, in nonreciprocal media, the
retrieved function G共xB,xA,t兲 Ⳮ G共xA,xB,ⳮt兲 is no longer time symmetric 共see Figure 6 for a 1D illustration兲. Interferometry in moving
media has potential applications in solar seismology and in infrasound 共Evers and Siegmund, 2009; Haney, 2009兲.
With similar simple modifications, global-scale interferometry
accounts for the Coriolis force of a rotating earth 共Ruigrok et al.,
2008兲 and electromagnetic interferometry accounts for nonreciprocal effects in bianisotropic media 共Slob and Wapenaar, 2009兲. A
moving conductive medium in the presence of a static magnetic field
is an example of a bianisotropic medium. Electromagnetic interferometry in bianisotropic media may find applications in controlledsource electromagnetic 共CSEM兲 acquisition with receivers in the air
in areas with strong tidal currents.
The wave equation for a medium with losses can be seen as a special case of the more general differential equation,
冋兺
n⳱1
⳵
n
an共x,t兲 ⴱ
Weaver 共2008兲 provides an alternative derivation based on Ward
identities. A recent extension 共Snieder et al., 2010兲 also accounts for
potential fields 共for which all an ⳱ 0 and H is independent of time in
equation 20兲.
Similarly, for a matrix-vector differential equation of the form
冋
A共x,t兲 ⴱ
册
D
Ⳮ Dx Ⳮ B共x,t兲 ⴱ u共x,t兲 ⳱ s共x,t兲,
Dt
共21兲
where A and B are medium-parameter matrices, D / Dt the material
time derivative, and Dx a spatial differential operator matrix, a unified Green’s matrix representation has been derived 共Wapenaar et
al., 2006兲. This representation, again consisting of a boundary and
a volume integral, captures interferometry for acoustic, elastodynamic, electromagnetic, poroelastic, piezoelectric, and electroseismic wave propagation as well as for diffusion and flow. For the situation of uncorrelated noise sources distributed along a boundary 共for
lossless media兲 or throughout a volume 共for media with losses兲, the
unified Green’s matrix representation is given by
兵G共xB,xA,t兲 Ⳮ Gt共xA,xB,ⳮ t兲其 ⴱ SN共t兲 ⬇ 具u共xB,t兲 ⴱ ut共xA,ⳮ t兲典
共22兲
共superscript t denotes transposition兲, where u共xA,t兲 and u共xB,t兲 are
the noise responses at xA and xB, respectively. In subscript notation,
this becomes
兵Gqp共xB,xA,t兲 Ⳮ G pq共xA,xB,ⳮ t兲其 ⴱ SN共t兲
⬇ 具uq共xB,t兲 ⴱ u p共xA,ⳮ t兲典.
共23兲
Note the resemblance to equation 10 for elastodynamic Green’s
function retrieval. For electroseismic waves, ut ⳱ 共Et,Ht,兵vs其t,ⳮ␶t1,
ⳮ␶t2,wt,p f 兲, where E and H are the electric and magnetic field vectors, vs the particle velocity of the solid phase, ␶i the traction, w the
filtration velocity of the fluid through the pores, and p f the pressure
of the fluid phase. Accordingly, for example, the 共9,1兲 element of
G共xB,xA,t兲, i.e., G9,1共xB,xA,t兲, is the vertical particle velocity of the
solid phase at xB from an impulsive horizontal electric current source
at xA. According to equations 22 and 23, it is retrieved by crosscorrelating the ninth element of u共xB,t兲, i.e., the vertical velocity noise
a)
Flow
xS
xA
xB
x
xS
b)
Unified formulations
N
75A219
⳵t
n ⳮ H共x,t兲 ⴱ
册
20
G(x B ,x A ,t) S N (t )
G(x A ,x B ,–t)S N (t)
15
10
5
u共x,t兲 ⳱ s共x,t兲,
共20兲
where an共x,t兲 are medium parameters, H共x,t兲 is a spatial differential
operator, and s共x,t兲 is a source function. Snieder et al. 共2007兲 derive
unified Green’s function representations for fields obeying this differential equation, assuming H共x,t兲 is either symmetric or antisymmetric. These representations, consisting of a boundary and a volume integral, capture interferometry for acoustic wave propagation
共with or without losses兲, diffusion, advection, bending waves in mechanical structures, and quantum mechanical scattering problems.
0
−5
−10
−1
–1.0
−0.8
−0.6
−0.4
–0.5
−0.2
0
0
0.2
0.4
0.6
0.5
0.8
1
t(s)
Figure 6. A 1D example of direct-wave interferometry in a moving
medium. 共a兲 Rightward- and leftward-propagating noise signals in a
rightward-flowing medium. 共b兲 Crosscorrelation of the responses at
xA and xB. The causal part stems from the rightward-propagating
wave and is interpreted as the Green’s function propagating “downwind” from xA to xB. The acausal part stems from the leftward-propagating wave and is interpreted as the time-reversed Green’s function
propagating “upwind” from xB to xA.
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75A220
Wapenaar et al.
field at xB with the first element of u共xA,t兲 being the horizontal electric noise field at xA 共Figure 7兲. For a further discussion on electroseismic interferometry, including numerical examples, see de
Ridder et al. 共2009兲.
Relation with the generalized optical theorem
It has recently been recognized 共Snieder et al., 2008; Halliday and
Curtis, 2009a兲 that the frequency-domain Green’s function representation for seismic interferometry resembles the generalized optical theorem 共Heisenberg, 1943; Glauber and Schomaker, 1953;
Newton, 1976; Marston, 2001兲, given by
ⳮ1
k
兵f共kA,kB兲 ⳮ f *共kB,kA兲其 ⳱
2j
4␲
冖
f共k,kB兲f *共k,kA兲d⍀ ,
共24兲
where f共kA,kB兲 is the far-field angle-dependent scattering amplitude
of a finite scatterer 共Figure 8兲, including all linear and nonlinear interactions of the wavefield with the scatterer.
The optical theorem has a form similar to interferometry representation 16 for acoustic waves. The analysis of this resemblance has
led to new insights in interferometry as well as in scattering theorems. Snieder et al. 共2008兲 use the generalized optical theorem to explain the cancellation of specific spurious arrivals in Green’s function extraction. Halliday and Curtis 共2009a兲 show that the generalized optical theorem can be derived from the interferometric Green’s
function representation and use this to derive an optical theorem for
surface waves in layered elastic media. Snieder et al. 共2009b兲 discuss
how the scattering amplitude can be derived from field fluctuations.
In other related work, Halliday and Curtis 共2009b兲 and Wapenaar et
al. 共2010兲 show that the Born approximation is an insufficient model
to explain all aspects of seismic interferometry, even for the situation
of a single point scatterer, and use this insight to derive improved
models for the scattering amplitude of a point scatterer.
Virtual receivers, reflectors, and imaging
Until now, we have discussed seismic interferometry as a method
that retrieves the response of a virtual source by crosscorrelating responses at two receivers. Using reciprocity, it is also possible to create a virtual receiver by crosscorrelating the responses of two sources. Curtis et al. 共2009兲 use this principle to turn earthquake sources
into virtual seismometers with which real seismograms can be recorded, located noninvasively deep within the earth’s subsurface.
They argue that this methodology has the potential to improve the
resolution of imaging the earth’s interior by earthquake seismology.
An earthquake source acts like a double couple; so by reciprocity, the
virtual receiver acts like a strainmeter, a device that is not easily implemented by a physical instrument. In a similar way, microseismic
sources near a reservoir could be turned into virtual receivers to improve the resolution of reservoir imaging 共Figure 9兲. Note that imaging using virtual receivers requires knowledge of the position of the
sources, but recording seismograms on the virtual seismometers
does not.
Another variant is the virtual reflector method 共Poletto and Farina,
2008; Poletto and Wapenaar, 2009兲. This method creates new seismic signals by processing real seismic responses of impulsive or
transient sources. Under proper recording coverage conditions, this
technique obtains seismograms as if there were an ideal reflector at
the position of the receivers 共or sources兲. The algorithm consists of
convolution of the recorded traces, followed by integration of the
crossconvolved signals along the receivers 共or sources兲. Similar to
other interferometry methods, the virtual reflector method does not
require information on the propagation velocity of the medium. Poletto and Farina 共2010兲 illustrate the method with synthetic marine
and real borehole data.
Curtis 共2009兲, Schuster 共2009, chapter 8兲, and Curtis and Halliday
共2010b兲 discuss source-receiver interferometry. This method combines the virtual-source and the virtual-receiver methodologies and
thus involves a double integration over sources and receivers. It creates the response of a virtual source observed by a virtual receiver.
This method is related to prestack redatuming 共Berryhill, 1984兲, in
which sources and receivers are repositioned from the acquisition
surface to a new datum plane in the subsurface, using one-way wavefield extrapolation operators based on a macromodel. In source-receiver interferometry, the operators are replaced by measured responses — for example, in VSPs. Hence, source-receiver interferometry can be seen as the data-driven variant of prestack redatuming.
Note, however, that in general the measured responses used in
source-receiver interferometry are full wavefields rather than oneway operators. Therefore, the application of source-receiver interferometry is not restricted to data-driven prestack redatuming, but it
can be used for other applications as well. For example, Halliday et
al. 共2010b兲 show that the elastodynamic version of source-receiver
interferometry can be seen as a generalization of a method that turns
Controlled transient sources
E1
v 3s
e
J1
v 3s
kB
kA
G9,1(x B ,x A ,t )
Uncorrelated noise sources
Figure 7. Principle of electroseismic interferometry for controlled
transient sources at the surface or uncorrelated noise sources in the
subsurface. In this example, the vertical component of the particle
velocity of the solid phase is crosscorrelated with the horizontal
component of the electric field, yielding the electroseismic response
of a horizontal electric current source observed by a vertical geophone.
Figure 8. The generalized optical theorem for the angle-dependent
scattering amplitude f共kA,kB兲 共Heisenberg, 1943兲 has a similar form
as the Green’s function representation for seismic interferometry.
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Tutorial on interferometry: Part 2
PP and PS data into SS data, previously proposed by Grechka and Tsvankin 共2002兲 and Grechka and Dewangan 共2003兲. In a similar fashion, the internal multiple prediction method of Jakubowicz 共1998兲
can be derived as a special case of source-receiver interferometry.
Also, the surface-wave-removal methods of Dong et al. 共2006兲, Curtis et al. 共2006兲, and Halliday et al. 共2007, 2010a兲 require physically
recorded and interferometrically constructed Green’s function estimates between the locations of an active source and active receiver.
Previously, the interferometric estimate was obtained by having to
place a receiver beside every source and turning the former into a virtual source 共or vice versa, using virtual-receiver interferometry兲.
However, by using source-receiver interferometry, this becomes unnecessary because the interferometric wavefield estimate can be
made between real source and real receiver directly 共Curtis and Halliday, 2010b兲.
Similar double integrals appear in the acoustic inverse scattering
imaging formulation of Oristaglio 共1989兲. Halliday and Curtis
共2010兲 derive explicitly a generalized version of Oristaglio’s formulation from a version of source-receiver interferometry for a medium
with scattering perturbations. The derivation was possible because
this form of interferometry is the first to combine active sources and
receivers, similar to geometries used for imaging.
Time-lapse seismic interferometry
As a consequence of the stability of time-reversed acoustics, seismic interferometry has large potential for time-lapse methods. We
have indicated the use of passive interferometry for monitoring
changes in volcanic interiors 共Sens-Schönfelder and Wegler, 2006;
Brenguier et al., 2008b兲. Using the same principles, Brenguier et al.
共2008a兲 monitor postseismic relaxation along the San Andreas fault
at Parkfield, California, U.S.A., and Ohmi et al. 共2008兲 monitor temporal variations of the crustal structure in the source region of the
2007 Noto Hanto earhquake in central Japan. Kraeva et al. 共2009兲
show a relation between seasonal variations of ambient noise crosscorrelations and remote microseismic activity related to ocean
storms, and Haney 共2009兲 reports on time-dependent effects in correlations of infrasound that arise from time-varying temperature
fields and temperature inversion layers in the atmosphere. The interpretation in all these methods is based on measuring the time shift in
either the direct wave or the coda wave of the Green’s functions retrieved by interferometry. These time shifts give information about
the average velocity change between the receivers, which can be further regionalized by tomographic inversion 共Brenguier et al.,
2008b兲.
In the field of controlled-source interferometry, Bakulin et al.
共2007兲 and Mehta et al. 共2008兲 discuss the potential of the virtualsource method for time-lapse reservoir monitoring. They exploit the
fact that virtual-source data are obtained from permanent downhole
or ocean-bottom-cable receivers and hence have a high degree of repeatability. Because virtual-source data represent reflection responses, local time-lapse changes in these data can be reliably attributed to local changes in the reservoir.
To better quantify the time-lapse changes in the data obtained by
seismic interferometry, the interferometric Green’s function representation 共equation 11兲 has been modified to account for time-lapse
changes according to
75A221
ˆ
Ĝ共xB,xA,␻ 兲 Ⳮ Ḡ*共xB,xA,␻ 兲
⳱ⳮ
冖
1
ˆ
共共⳵iĜ共xB,x,␻ 兲兲Ḡ*共xA,x,␻ 兲
⳵D j ␻␳ 共x兲
ˆ
ⳮ Ĝ共xB,x,␻ 兲⳵iḠ*共xA,x,␻ 兲兲nid2x
Ⳮ j␻
冕
ˆ
⌬␬ˆ 共x,␻ 兲Ĝ共xB,x,␻ 兲Ḡ*共xA,x,␻ 兲d3x,
共25兲
D
¯ˆ *共x,␻ 兲 共Vasconcelos and Snieder,
with ⌬␬ˆ 共x,␻ 兲 ⳱ ␬ˆ 共x,␻ 兲 ⳮ ␬
2008a; Douma, 2009; Vasconcelos et al., 2009兲. Here, the quantities
with/without a bar refer to the reference/monitor state 共for simplicity, we assume that time-lapse changes occur only in the compressibility兲. The equivalent theory for source-receiver interferometry is
given in Halliday and Curtis 共2010兲. Equation 25 and its generalization for other wave types 共Wapenaar, 2007兲 provides a basis for deriving local time-lapse changes of the medium parameters from interferometric time-lapse data. This is the subject of ongoing research.
Interferometry by deconvolution and crosscoherence
In the previous treatment of interferometry, we focused on
Green’s function extraction by crosscorrelation. Time reversal corresponds to complex conjugation in the frequency domain, so the
crosscorrelation is, in the frequency domain, given by
Ĉ共xB,xA,␻ 兲 ⳱ û共xB,␻ 兲û*共xA,␻ 兲.
共26兲
According to expression 19, the crosscorrelation does not just give
the superposition of the Green’s function and its time-reversed counterpart because the left-hand side of that expression is convolved
with the autocorrelation of the noise that excites the field fluctuations. This means that equation 26 gives the product of the Green’s
function and the power spectrum ŜN共 ␻ 兲 of the noise. The power
spectrum thus leaves an imprint on the extracted Green’s function
unless it is properly accounted for. This imprint can be eliminated by
using deconvolution instead of crosscorrelation. In the frequency
domain, deconvolution corresponds to spectral division; hence, the
deconvolution approach consists of replacing expression 26 by
x
u (x,x A ,t)
x A ( Virtual receiver)
Complex
overburden
xB
G(x A ,xB , t)
u(x,x B ,t)
Simpler
middle
overburden
Target
Figure 9. Using reciprocity, Bakulin and Calvert’s virtual-source
method 共Figure 2兲 can be reformulated into a virtual receiver method. Receivers at the surface record the direct and the reflection responses of microseismic sources above a deeper target. Crosscorrelation and summing over receiver locations gives the reflection response at a virtual receiver at the position of a microseismic source,
free of overburden distortions.
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75A222
Wapenaar et al.
D̂共xB,xA,␻ 兲 ⳱
û共xB,␻ 兲
û共xA,␻ 兲
共27兲
.
When û共xA,␻ 兲 is small, this spectral division is unstable. In practice,
one needs to regularize the deconvolution. The simplest way to do
this is to use the following water-level regularization:
D̂共xB,xA,␻ 兲 ⳱
û共xB,␻ 兲û*共xA,␻ 兲
兩û共xA,␻ 兲兩2 Ⳮ ␧2
,
共28兲
where ␧2 is a stabilization parameter. When ␧2 ⳱ 0, expression 28 reduces to equation 27; for ␧2 兩û共xA,␻ 兲兩2, equation 28 corresponds to
a scaled version of the correlation defined in expression 26.
A significant difference between crosscorrelation and deconvolution is that crosscorrelation gives the Green’s function but deconvolution does not. This raises the question, what wave state is retrieved
by deconvolving field measurements recorded at different points?
There is a simple proof that the wave states obtained by crosscorrelation, deconvolution, and regularized deconvolution all satisfy the
same equation as the real system does 共Snieder et al., 2006a兲. Let us
denote the field equation of the system by
L̂共x,␻ 兲û共x,␻ 兲 ⳱ 0.
共29兲
For the acoustic wave equation in a constant-density medium, for
example, the operator L̂ is given by L̂共x,␻ 兲 ⳱ ⵜ2 Ⳮ ␻ 2 / c2共x兲. The
right-hand side of expression 29 equals zero, so this expression
holds for source-free regions, which is the case at the receivers. Applying L̂ to equation 27 with xB replaced by x gives
L̂共x,␻ 兲D̂共x,xA,␻ 兲 ⳱ L̂共x,␻ 兲
⳱
冉
1
û共xA,␻ 兲
û共x,␻ 兲
û共xA,␻ 兲
冊
L̂共x,␻ 兲û共x,␻ 兲 ⳱ 0, 共30兲
where in the second identity we used that L̂共x,␻ 兲 acts on the
x-coordinates only and where the field equation 29 is used in the last
identity. The same reasoning applies to the correlation of expression
26 and the regularized deconvolution in expression 28. All of these
procedures thus produce a wave state that satisfies the same wave
equation as the original system does. For the correlation, this wave
state is the Green’s function; but for the deconvolution, a different
wave state is obtained.
To understand which wave state is extracted by deconvolution, we
note that
D̂共xA,xA,␻ 兲 ⳱
û共xA,␻ 兲
û共xA,␻ 兲
⳱ 1.
共31兲
This corresponds, in the time domain, to
D共xA,xA,t兲 ⳱ ␦ 共t兲.
共32兲
Deconvolution thus gives a wave state that, for t ⫽ 0, vanishes at the
virtual-source location xA. This means that the wavefield vanishes at
that location, and hence, the phrase “clamped boundary condition”
has been used 共Vasconcelos and Snieder, 2008a兲. Deconvolution
thus gives a wave state where the field vanishes at one point in space.
This wave state is, in general, not equal to the Green’s function.
Despite this strange boundary condition, interferometry by deconvolution has a distinct advantage for attenuating media. Consider
the example of Figure 1a of Part 1 of this tutorial where a plane wave
propagates along a line from a source at xS to receivers at xA and xB,
respectively. For a homogeneous attenuating medium, the field
recorded at xA equals û共xA,␻ 兲 ⳱ Ĝ共xA,xS,␻ 兲N̂共 ␻ 兲 ⳱ exp共ⳮ␥ 共xA
ⳮ xS兲兲exp共ⳮjk共xA ⳮ xS兲兲N̂共 ␻ 兲, where ␥ is an attenuation coefficient and N̂共 ␻ 兲 is the source spectrum.Asimilar expression holds for
the field at xB. The correlation of the fields recorded at xA and xB is
given by
Ĉ共xB,xA,␻ 兲 ⳱ eⳮ␥ 共xAⳭxBⳮ2xS兲eⳮjk共xBⳮxA兲ŜN共␻ 兲, 共33兲
with ŜN共 ␻ 兲 ⳱ 兩N̂共 ␻ 兲兩2. This field has the same phase as the field that
propagates from xA to xB, but the attenuation is incorrect because it
depends on the source location xS, which is, of course, not related to
the field that propagates between xA and xB. In contrast, the deconvolution of the recorded fields satisfies
D̂共xB,xA,␻ 兲 ⳱ eⳮ␥ 共xBⳮxA兲eⳮjk共xBⳮxA兲 ⳱ Ĝ共xB,xA,␻ 兲,
共34兲
which does correctly account for the phase and the amplitude and
which does not depend on N̂共 ␻ 兲. This property of the deconvolution
approach for 1D systems has been used to extract the velocity and attenuation in the near surface 共Trampert et al., 1993; Mehta et al.,
2007b兲 and to determine the structural response of buildings from incoherent ground motion 共Snieder and Şafak, 2006; Thompson and
Snieder, 2006; Kohler et al., 2007兲. The deconvolution method has
even been used to detect changes in the near-surface shear-wave velocity during shaking caused by an earthquake 共Sawazaki et al.,
2009兲.
The application of deconvolution interferometry changes when
one can separate the wavefield into an unperturbed wave u0 and a
perturbation uS 共Vasconcelos and Snieder, 2008b兲. Such a separation
can be achieved by time gating when impulsive shots are used
共Bakulin et al., 2007; Mehta et al., 2007a兲, by using array methods,
or by using four-component data. In this case, one can define a new
deconvolution,
D̂⬘共xB,xA,␻ 兲 ⳱
ûS共xB,␻ 兲
û0共xA,␻ 兲
共35兲
,
which gives an estimate of the perturbed Green’s function ĜS. This
modified deconvolution method has been used to illuminate the San
Andreas fault from the side using drill-bit noise 共Figure 10兲 and for
subsalt imaging from below using internal multiples 共Vasconcelos et
al., 2008兲. A comparison of crosscorrelation, deconvolution, and
multidimensional deconvolution 共presented in the next section兲 is
given by Snieder et al. 共2009a兲.
A method related to deconvolution is crosscoherence, defined as
Ĥ共xB,xA,␻ 兲 ⳱
û共xB,␻ 兲 û*共xA,␻ 兲
兩û共xB,␻ 兲兩 兩û共xA,␻ 兲兩
.
共36兲
The crosscoherence can be seen as a spectrally normalized crosscorrelation or as a variant of deconvolution that is symmetric in û共xA,␻ 兲
and û共xB,␻ 兲. This method of combining data is proposed by Aki in
his seminal papers on retrieving surface waves from microtremors
共1957, 1965兲. It has been used extensively in engineering 共Bendat
and Piersol, 2000兲 in the extraction of response functions and is commonly used to determine shallow shear velocity from ground vibrations, e.g., Chávez-García and Luzón 共2005兲. Note that the reason-
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Tutorial on interferometry: Part 2
ing leading to equation 30 is not applicable to the crosscoherence because of the presence of the normalized spectrum in the denominator
of expression 36. Hence, the crosscoherence does not necessarily
lead to a wave state that satisfies the same equation as the real system
does.
75A223
Ⳮ
⬘ ,x共i兲
C共xB,xA⬘ ,t兲 ⳱ 兺 uⳮ共xB,x共i兲
S ,t兲 ⴱ u 共xA
S ,ⳮ t兲
共41兲
i
and
Ⳮ
⬘ ,x共i兲
⌫ 共xA,xA⬘ ,t兲 ⳱ 兺 uⳭ共xA,x共i兲
S ,t兲 ⴱ u 共xA
S ,ⳮ t兲. 共42兲
i
Interferometry by multidimensional deconvolution
Interferometry by multidimensional deconvolution 共MDD兲 is the
natural extension of interferometry by deconvolution to two or three
dimensions. It has been proposed for controlled-source data
共Schuster and Zhou, 2006; Wapenaar et al., 2008a兲 as well as for passive data 共Wapenaar et al., 2008b兲. Here, we discuss the principle for
controlled-source data and briefly indicate the modifications for
noise data.
Consider again Figure 2, which we initially used to introduce the
virtual-source method of Bakulin and Calvert 共2004兲. We express
the upgoing wavefield at xB as
冕
G共xB,xA,t兲ⴱuⳭ共xA,x共i兲
S ,t兲dxA, 共37兲
where the plus and minus superscripts refer to downgoing and upgoing waves, respectively. Note that the integration takes place along
the receivers at xA in the borehole. This convolutional data representation is valid in media with or without losses. However, unlike
equation 1, which is an explicit but approximate expression for the
Green’s function G共xB,xA,t兲 共convolved with the autocorrelation
Ss共t兲 of the source wavelet兲, equation 37 is an implicit but exact expression for G共xB,xA,t兲 共with G共xB,xA,t兲 being the reflection response of the medium below the receiver level with a homogeneous
half-space above it 关Wapenaar, Slob, et al., 2008兴兲. Equation 37 can
be solved by MDD, assuming responses are available for many
source positions x共i兲
S . In that case, equation 37 holds for each source
separately. In the frequency domain, the resulting set of simultaneous equations can be represented in matrix notation 共Berkhout,
1982兲 according to
Ûⳮ ⳱ ĜÛⳭ,
0
共38兲
共i兲
where the 共 j,i兲 element of ÛⳭ is given by ûⳭ共x共j兲
A ,x S , ␻ 兲, etc. Equation 38 can be solved for Ĝ, e.g., via weighted least-squares inversion 共Menke, 1989兲, according to
Ĝ ⳱ ÛⳮW兵ÛⳭ其†共ÛⳭW兵ÛⳭ其† Ⳮ ␧2I兲ⳮ1,
with
冕
G共xB,xA,t兲 ⴱ ⌫ 共xA,xA⬘ ,t兲dxA,
4
共40兲
3
c
1
2
safz
1
a
2
?
b
共39兲
where the dagger denotes transposition and complex conjugation, W
is a diagonal weighting matrix, I is the identity matrix, and ␧2 is a stabilization parameter. Equation 39 is the multidimensional extension
of equation 28. Applying this equation for each frequency component and transforming the result to the time domain accomplishes interferometry by MDD.
To get more insight into equation 37 and its solution by MDD, we
convolve both sides with the time-reversed downgoing wavefield
共i兲
uⳭ共x⬘A,x共i兲
S ,ⳮt兲 and sum over the source positions x S 共van der Neut et
al., 2010兲. This gives
C共xB,xA⬘ ,t兲 ⳱
Interferometric image
SAFOD
Depth (km)
uⳮ共xB,x共i兲
S ,t兲 ⳱
Note that, according to equation 41, C共xB,x⬘A,t兲 is nearly identical to
the correlation function of equation 1; hence, equation 41 represents
the virtual-source method of Bakulin and Calvert 共2004, 2006兲 but
applied to decomposed wavefields 共Mehta et al., 2007a兲. According
to equation 42, ⌫ 共xA,x⬘A,t兲 contains the correlation of the incident
wavefields. We call this the point-spread function. For equidistant
sources and a homogeneous overburden, the point-spread function
will approach ⌫ 共xA,x⬘A,t兲 ⳱ ␦ 共xA ⳮ x⬘A兲Ss共t兲 共with xA and x⬘A both in
the borehole兲. Hence, for this situation, equation 40 reduces to
C共xB,x⬘A,t兲 ⳱ G共xB,x⬘A,t兲 ⴱ Ss共t兲, meaning that for this situation, the
correlation method gives the correct Green’s function, convolved
with Ss共t兲.
For the situation of an irregular source distribution and/or a complex overburden, the point-spread function can become a complicated function of space and time. Equation 40 shows that the correlation
method 共i.e., Bakulin and Calvert’s virtual-source method兲 gives the
Green’s function, distorted by the point-spread function. These distortions manifest themselves as an irregular radiation pattern of the
virtual source and artifacts 共spurious multiples兲 related to the onesided illumination. The true Green’s function follows by multidimensionally deconvolving the correlation function by the pointspread function. Van der Neut and Bakulin 共2009兲 demonstrate that
this indeed improves the radiation pattern of the virtual source and
suppresses the artifacts.
Note that MDD can be carried out without knowing the source po-
3
?
?
4
–3
–2
–1
0
Distance from SAF (km)
1
Figure 10. Images of the San Andreas Fault Observatory at depth
共SAFOD兲. An image from deconvolution interferometry using drillbit noise 共Vasconcelos and Snieder, 2008b兲 is superimposed on an
image obtained from surface seismic data and microseismic events
from the San Andreas fault zone 共SAFZ兲, measured at the surface
and in the pilot hole 共Chavarria et al., 2003兲. Event 2 is a prominent
reflector, consistent with the surface trace of the SAF. Event 3 is interpreted to be a blind fault at Parkfield. Events 1 and 4 are interpreted to be artifacts, possibly because of drillstring multiples and improperly handled converted-wave modes.
Downloaded 15 Sep 2010 to 138.67.20.77. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/
75A224
Wapenaar et al.
sitions and the medium parameters 共similar to crosscorrelation interferometry兲 and without making assumptions about the regularity of
the source positions x共i兲
S and the attenuation parameters of the medium 共the latter properties are unique for the deconvolution approach兲.
The application of equations 41 and 42 requires decomposition into
downgoing and upgoing waves and hence the availability of pressure and particle velocity data. The retrievable source-receiver offset range by MDD is limited by the highest velocity in the domain
between the sources and the receivers. The available spatial bandwidth in the recorded data may not always be sufficient to retrieve
full-range offsets. This is likely to occur in alternating velocity
zones. It also occurs in areas where velocities decrease with increasing depth, which is the usual situation for electromagnetic waves
共Slob, 2009兲.
For the situation of uncorrelated noise sources, equations 41 and
42 would need to be replaced by C共xB,x⬘A,t兲 ⳱ 具uⳮ共xB,t兲 ⴱ uⳭ共x⬘A,
ⳮt兲典 and ⌫ 共xA,x⬘A,t兲 ⳱ 具uⳭ共xA,t兲 ⴱ uⳭ共x⬘A,ⳮt兲典, analogous to equation 19. For a further discussion of MDD applied to passive data, see
Wapenaar et al. 共2008b兲; van Groenestijn and Verschuur 共2009兲; and
van der Neut et al. 共2010兲.
The MDD principle is not entirely new. For example, it has been
used for multiple elimination from ocean bottom data 共Wapenaar
and Verschuur, 1996; Amundsen, 1999; Holvik and Amundsen,
2005兲. Like the 1D deconvolution method of Snieder et al. 共2006a兲
discussed above, this can be seen as a methodology that changes the
boundary conditions of the system; it transforms the response of the
subsurface, including the reflecting ocean bottom and water surface,
into the response of a subsurface without these reflecting boundaries. In hindsight, this methodology appears to be an extension of a
1D deconvolution approach proposed by Riley and Claerbout
共1976兲. Slob et al. 共2007b兲 apply MDD to up/down decomposed
CSEM data 共Amundsen et al., 2006兲 and demonstrate the insensitivity to dissipation as well as the effect of changing the boundary conditions: The effect of the air wave, a notorious problem in CSEM
prospecting, is largely suppressed.
Interferometry by MDD is, from a theoretical point of view, more
accurate than the crosscorrelation approach; however, the involved
processing is less attractive because it is not a trace-by-trace process
but involves inversion of large matrices. Moreover, in most cases it
requires decomposition into downgoing and upgoing fields. Nevertheless, the fact that interferometry by MDD corrects for an irregular
source distribution, suppresses spurious multiples due to one-sided
illumination, improves the radiation pattern of the virtual source,
and accounts for dissipation makes it a worthwhile method to be investigated as an alternative to interferometry by crosscorrelation for
passive and controlled-source data applications.
CONCLUSIONS
In Part 1, we discussed the basic principles of seismic interferometry in a heuristic way. In this second part, we have discussed interferometry in a more formal way. First, we reviewed the methodology
of time-reversed acoustics, pioneered by Mathias Fink and coworkers, and used physical arguments ofArnaud Derode to derive seismic
interferometry from the principle of time-reversed acoustics. We
continued with a mathematical derivation based on general reciprocity theory leading to exact Green’s function representations, which
are the basis for controlled-source as well as passive interferometry.
Finally, we discussed generalizations and variations of these repre-
sentations and showed that those generalizations form the basis for a
rich variety of new applications.
The fact that seismic interferometry leads to new responses directly from measured data has stirred a lot of enthusiasm and cooperation between researchers in seismology, acoustics, and electromagnetic prospecting in the past decennium. We believe we have only
seen the start and expect to see many new developments and applications in different fields in the years to come.
ACKNOWLEDGMENTS
This work is supported by the Netherlands Research Centre for Integrated Solid Earth Science 共ISES兲, the Dutch Technology Foundation 共STW兲 共grant DCB.7913兲, and the U. S. National Science Foundation 共NSF兲 共grant EAS-0609595兲. We thank associate editor Sven
Treitel and reviewers Dirk-Jan van Manen, Matt Haney, and Huub
Douma for their valuable comments and suggestions, which improved this paper. We are grateful to Mathias Fink, David Halliday,
and Ivan Vasconcelos for permission to show the time-reversed
acoustics experiment, the interferometric ground-roll removal experiment, and the interferometric SAF image, respectively.
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